Prestack phase-shift migration of separate offsets

Prestack phase-shift migration is implemented by evaluating the offset-wavenumber (k(h)) integral using the stationary-phase method. Thus, the stationary point along k(h) must be calculated prior to applying the phase shift. This type of implementation allows for migration of separate offsets, as opposed to migration of the whole prestack data when using the original formulas. For zero-offset data, the stationary point (k(h) = 0) is known in advance, and, therefore, the phase-shift migration can be implemented directly. For nonzero-offset data, we first evaluate the k(h) that corresponds to the stationary point solution either numerically or through analytical approximations. The insensitivity of the phase to k(h) around the stationary point solution (its maximum) implies that even an imperfect k(h) obtained analytically can go a long way to getting an accurate image. In transversely isotropic media, the analytical solutions of the stationary point (k(h)) include more approximations than those corresponding to isotropic media (i.e., approximations corresponding to weaker anisotropy). Nevertheless, the resultant equations, obtained using Shanks transform and perturbation theory, produce accurate migration signatures for strong anisotropy (eta approximate to 0.3) and even large offset-to-depth ratios (>2). The analytical solutions are particularly accurate in predicting the nonhyperbolic moveout behavior associated with anisotropic media, a key ingredient to performing an accurate nonhyperbolic moveout inversion for strongly anisotropic media. Although the prestack correction achieved using the phase-shift method can also be obtained using a cascade of NMO correction, dip-moveout (DMO) correction, and zero-offset time migration, the prestack approach can handle sharper velocity models more efficiently. In addition, the resulting operator is sharper then that obtained from the DMO method. Synthetic, including the Marmousi, data applications of the proposed prestack migration demonstrate its usefulness.